The Research on precise estimate for the regularity of diffusion Operator of diffusion process with absorbed boundary condition and Its application

我们研究了与具有吸收边界(Dirichlet边界)的扩散过程有关的扩散方程的解的性质和数值计算方法。首先，我们研究了具有Dirichlet边界条件的扩散算子的光滑性的精确估计，得到了与没有边界条件的扩散算子的估计类似的估计。然后研究了具有吸收边界的扩散过程期望的数值计算方法。我们研究了一种类似于无边界条件下的KLNV方法，并得到了三阶近似公式。遗憾的是，公式中出现了惠塔克函数，实现困难，无法进行计算机模拟。

We studied the property and numerical computation methods for solutions of diffusion equation related to diffusion processes with absorbed boundary (Dirichlet boundary) which appears in pricing knock-out derivatives. First, we studied about precise estimates of smoothness for diffusion operators with Dirichlet boundary condition, and obtained similar estimates to estimates in the case of diffusion operators without boundary conditions. Then we studied numerical computation methods for the expectation of diffusion process with an absorbed boundary. We studied a similar method to KLNV method in the case of no boundary conditions, and obtained 3rdorder approximation formula. Unfortunately Whitakker function appears in the formula,the implementation is difficult, and so we could not do the computer simulation.